What i feel is that Gobekli Tepe was never a Residential Area like a cave.
But rather it was a religious and ceremonial structure.
Harsh, it is not a single construction.
It might have started from bare form structure that the Hunter gatherer community knew how to construct at their times.
As you are aware, Gobekli Tepe was buried periodically. (for what reasons it is never clear)
refer to this
link for the dating of the layers.
Advances in culture means advances in technology. Their building methods advanced. The structure at each layer became more elaborate and sophisticated.
So, when you ask why and how a "hunter-gatherer Culture' did build Gobekli Tepe, i would ask you - Why did ancient Indians buied the temple, or ancient Christians build churches and basilicas?
Because they could and because they wanted to.
Religion is always a moving force for the masses.
When Ancient Humans started worship forces of natures as God and all, they built abodes to them, thinking the gods will reside there.
The religious place became the community's social communing centre too.
matters of law and commerce relating to the community were discussed there.
Certain Communities developed into the major centres. The ceremonial/religious centres of these major communities became the major religious centres and places oif pilgrimage.
And please note that a "Hunter-gather" culture wont stay Hunter-gather-ish.
No culture sprang into life as advanced. They would have started as Hunter-gatherers and would surely advance through into settled, governed and modern civilizations.
Likewise, do not consider that the people who built Gobekli Tepe would have stayed Hunting & gathering all the time till the last burial of Gobekli Tepe.